Dinaciclib, Bortezomib, and Dexamethasone in Treating Patients With Relapsed Multiple Myeloma

Inclusion Criteria:

* Serum creatinine =\< 2.5 mg/dL
* Absolute neutrophil count \>= 1000/uL
* Untransfused platelet count \>= 75000/uL
* Hemoglobin \>= 8 g/dL
* Total bilirubin =\< 1.5 times institutional upper limit of normal (ULN)
* Patients with relapsed multiple myeloma who have already received one or more standard treatment regimens
* Measurable disease of multiple myeloma as defined by at least ONE of the following:

  * Serum monoclonal protein \>= 1 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
  * Monoclonal bone marrow plasmacytosis \>= 30% (evaluable disease)
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2
* Adequate residual organ function per treating physician discretion; Note: there is no limit with regard to the number of prior therapies
* Provide informed written consent
* Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only
* Willing to provide samples for correlative research purposes
* Willing to return to consenting institution for follow-up during the study
* Recovered (i.e., =\< grade 1 toxicity) from the reversible effects of prior antineoplastic therapy

Exclusion Criteria:

* Any of the following recent therapies:

  * Alkylators (e.g. melphalan, cyclophosphamide) =\< 14 days prior to registration
  * Anthracyclines =\< 14 days prior to registration
  * High dose corticosteroids, immune modulatory drugs (thalidomide or lenalidomide), or proteosome inhibitors (bortezomib) =\< 7 days prior to registration
* Concomitant high dose corticosteroids (concurrent use of corticosteroids); EXCEPTION: patients may be on chronic steroids (maximum dose 20 mg/day prednisone equivalent) if they are being given for disorders other than amyloid, i.e., adrenal insufficiency, rheumatoid arthritis, etc.
* Other active malignancy =\< 2 years prior to registration; EXCEPTIONS: non-melanotic skin cancer or carcinoma in-situ of the cervix; NOTE: if there is a history of prior malignancy, they must not be receiving other specific treatment for their cancer
* Any of the following:

  * Pregnant women or women of reproductive ability who are unwilling to use 2 effective methods of contraception from the time of signing the informed consent form through 30 days after the last dose of study drug
  * Nursing women
  * Men who are unwilling to use a condom (even if they have undergone a prior vasectomy) while having intercourse with any woman, while taking the drug and for 30 days after stopping treatment
* 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, radiotherapy, or any ancillary therapy considered investigational; NOTE: bisphosphonates are considered to be supportive care rather than therapy, and are thus allowed while on protocol treatment
* Peripheral neuropathy \>= grade 2 on clinical examination during the screening period
* Major surgery =\< 14 days prior to registration
* Currently taking strong or moderate inhibitors/inducers of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4); Note: dinaciclib is a CYP3A4 substrate; patients should not take grapefruit/grapefruit juice or St. John's wort; use of strong or moderate CYP3A4 inhibitors is prohibited from \< 7 days prior to registration; use of CYP3A4 inducers is prohibited from =\< 7 days prior to registration
* Any of the following conditions:

  * Myocardial infarction =\< 6 months prior to registration or has New York Heart Association (NYHA) class III or IV heart failure
  * Uncontrolled angina
  * Severe uncontrolled ventricular arrhythmias
  * Electrocardiographic evidence of acute ischemia
  * Active conduction system abnormalities; NOTE: prior to study entry, any electrocardiogram (ECG) abnormality at screening must be documented by the investigator as not medically relevant
* Known hypersensitivity to bortezomib, boron, or mannitol
* Serious medical or psychiatric illness likely to interfere with participation in this clinical study per the judgment of the treating physician