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