Carfilzomib and Dexamethasone in Treating Patients With Multiple Myeloma Who Previously Underwent a Stem Cell Transplant

Inclusion Criteria:

* Creatinine =\< 3 mg/dL
* Absolute neutrophil count \>= 1,000/uL
* Platelet count \>= 75,000/uL
* Hemoglobin \>= 8.0 g/dL
* Previous diagnosis of symptomatic multiple myeloma (MM)
* Received single autologous stem cell transplantation 60-120 days prior to registration
* Received the autologous SCT =\< 12 months of their diagnosis of myeloma to be eligible for the study
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2
* Recovered from toxicity of previous chemotherapy (excludes grade 1 neurotoxicity and hematological toxicity)
* Voluntary written informed consent before performance of any study-related procedure not part of normal medical care, with the understanding that the subject may withdraw consent at any time without prejudice to future medical care
* Negative pregnancy test performed =\< 7 days prior to registration, for women of childbearing potential only
* Willingness to return to one of the enrolling institutions for follow-up (during the active monitoring phase of the study); NOTE: during the active monitoring phase of a study (i.e., active treatment and observation), participants must be willing to return to the consenting institution for follow-up
* Measurable disease of multiple myeloma at the time of baseline values for disease assessment 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
  * NOTE: for patients with no relapse prior to transplant, measurable disease at the time of diagnosis
  * NOTE: for patients who have had a disease relapse prior to transplant, measurable disease at the time of the most recent relapse immediately prior to transplant; NOTE: if the patient had treatment for the relapsed disease prior to transplant, the patient must have measurable disease at the time of relapse prior to this therapy
* Willing to provide bone marrow and blood samples for correlative research purposes

Exclusion Criteria:

* Prior allogeneic bone marrow/peripheral blood stem cell transplant
* Evidence of disease progression post SCT at the time of consideration for the study enrollment
* Myocardial infarction =\< 6 months prior to registration
* New York Heart Association (NYHA) class III or IV heart failure
* Uncontrolled angina
* Severe uncontrolled ventricular arrhythmias
* Electrocardiographic (ECG) evidence of acute ischemia or active conduction system abnormalities; NOTE: prior to study entry, any ECG abnormality at screening has to be documented by the investigator as not medically relevant
* Seroreactivity for human immunodeficiency virus (HIV), human T-cell lymphotrophic virus (HTLV) I or II, hepatitis B virus (HBV), or hepatitis C virus (HCV)
* Other active malignancy requiring therapy; EXCEPTIONS: non-melanotic skin cancer or carcinoma-in-situ of the cervix; NOTE: if there is a history or prior malignancy, they must not be receiving other specific treatment for their cancer
* Pregnant women or women of reproductive capability who are unwilling to use effective contraception
* 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 28 days after stopping treatment
* Other co-morbidity, which would interfere with patient's ability to participate in the trial, e.g. uncontrolled infection, uncompensated lung disease
* 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
* Known allergies to any of the components of the investigational treatment regimen or required ancillary treatments