Pembrolizumab, Ixazomib Citrate, and Dexamethasone in Treating Patients With Relapsed Multiple Myeloma

Inclusion Criteria:

* Diagnosis of relapsed, symptomatic multiple myeloma by International Myeloma Working Group (IMWG) diagnostic criteria for multiple myeloma
* Age \>= 18 years
* Calculated creatinine clearance (using Cockcroft-Gault equation below) \>= 30 mL/min (obtained =\< 14 days prior to registration)
* Absolute neutrophil count (ANC) \>= 1000/mm\^3 (obtained =\< 14 days prior to registration)
* Platelet count \>= 75000/mm\^3; Note: Platelet transfusion is not allowed =\< 3 days prior to registration (obtained =\< 14 days prior to registration)
* Hemoglobin \>= 8.0 g/dL (obtained =\< 14 days prior to registration)
* Total bilirubin =\< 1.5 x upper limit of normal (ULN) (obtained =\< 14 days prior to registration)
* Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN (obtained =\< 14 days prior to registration)
* Must have relapsed or refractory disease after treatments including three therapies: proteasome inhibitors, immunomodulatory imide drugs (IMiDs), and anti-CD38 antibody
* Measurable disease of multiple myeloma as defined by at least ONE of the following:

  * Serum monoclonal protein \>= 1.0 g/dL
  * \>= 200 mg of monoclonal protein in the urine on 24 hour electrophoresis
  * Serum immunoglobulin free light chain \>= 10 mg/dL AND abnormal serum immunoglobulin kappa to lambda free light chain ratio
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, or 1
* Provide informed written consent
* Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only
* Willing to follow strict birth control measures;

  * Female patients: If they are of childbearing potential, agree to one of the following:

    * Practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent form through 120 days after the last dose of study drug, AND must also adhere to the guidelines of any treatment-specific pregnancy prevention program, if applicable, OR
    * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject; (periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception)
  * Male patients: even if surgically sterilized (i.e., status post-vasectomy), must agree to one of the following:

    * Agree to practice effective barrier contraception during the entire study treatment period and through 120 days after the last dose of study drug, OR
    * Must also adhere to the guidelines of any treatment-specific pregnancy prevention program, if applicable, OR
    * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject; (periodic abstinence (e.g., calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception)
* Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study)
* Willing to provide bone marrow and blood samples for planned research

Exclusion Criteria:

* Myeloma disease that is refractory to ixazomib treatment
* Has a known additional malignancy that is progressing or requires active treatment; exceptions include early stage cancers (carcinoma in situ or stage 1) treated with curative intent, basal cell carcinoma of the skin, squamous cell carcinoma of the skin, in situ cervical cancer, or in situ breast cancer that has undergone potentially curative therapy
* Any of the following:

  * Pregnant women
  * Nursing women
  * Men or women of childbearing potential who are unwilling to employ adequate contraception
* Other co-morbidity which would interfere with patient's ability to participate in trial, e.g. uncontrolled infection, uncompensated heart or lung disease
* Other concurrent chemotherapy, or any ancillary therapy considered investigational =\< 14 days prior to study registration; NOTE: Bisphosphonates are considered to be supportive care rather than therapy, and are thus allowed while on protocol treatment
* Peripheral neuropathy \>= grade 3 on clinical examination or grade 2 with pain during the screening period
* Major surgery =\< 14 days prior to study registration
* Radiotherapy =\< 14 days prior to registration; NOTE: If the involved field is small, 7 days will be considered a sufficient interval between treatment and administration of study drugs
* Participation in any other clinical trials with other investigational agents not included in this trial =\< 21 days prior to registration
* Has active autoimmune disease that has required systemic treatment =\< 2 years prior to study registration (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs);

  * NOTE: Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment
* Has a history of (non-infectious) pneumonitis that required steroids or current pneumonitis
* Has a known history of interstitial lung disease
* Has an active infection requiring systemic therapy
* Has a history of current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator
* Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial
* Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment
* Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent
* Has a known history of human immunodeficiency virus (HIV) (HIV 1/2 antibodies)
* Has known active hepatitis B (e.g., hepatitis B surface antigen \[HBsAg\] reactive) or hepatitis C (e.g., hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] \[qualitative\] is detected)
* Has a known history of active TB (Bacillus tuberculosis)
* Has received a live vaccine =\< 30 days prior to study registration
* Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis
* Allogeneic hematopoietic stem cell transplant
* Systemic treatment with strong CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of St. John's wort =\< 14 days prior to registration
* Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of ixazomib including difficulty swallowing