First-line Pomalidomide, Bortezomib, and Dexamethasone For AL Amyloidosis or LCDD

Inclusion Criteria: * Understand and voluntarily sign an informed consent form * Able to adhere to protocol requirements * Histologically confirmed AL or LCDD (any time prior to screening) * Up to 1 cycle of prior therapy allowed (maximum of 120 mg total dexamethasone (or equivalent amount of prednisone), 4 days of melphalan, and/or 4 doses of velcade; at least 4 wks has to have had passed since last dose of melphalan, 2 wks since last velcade or glucocorticoid dose * Measurable light chain elevation, as defined by: * A difference between the involved immunoglobulin free light chain and uninvolved light chain and uninvolved light chain (dFLC) of \>= 5 mg/dL AND abnormal serum immunoglobulin kappa lambda free light chain ratio * EXCEPTION: during the DOSE ESCALATION PORTION of the study only, a measurable M-protein (\>= 0.5 g/dL) on serum protein electrophoresis (SPEP) or a measurable urinary light chain (\>= 200 mg/24 hrs) by urine protein electrophoresis (UPEP) without a dFLC meeting the above criteria is acceptable; subjects without a dFLC \>= 5 mg/dL treated at the MTD will not count towards the expansion cohort * Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 at study entry * Demonstrated clonal population of plasma cells in the bone marrow or positive immunohistochemical stain with anti-light chain anti-sera of amyloid fibrils * NTproBNP \< 8500 pg/mL * Absolute neutrophil count \>= 1000/mm\^3 * Platelet count \>= 75,000/mm\^3 * Serum creatinine =\< 2.5 mg/dL * Total bilirubin =\< 1.5 mg/dL * AST(SGOT) and ALT(SGPT) =\< 3 x upper limit of normal (ULN) * All study participants must be registered into the mandatory POMALYST REMS™ program, and be willing and able to comply with the requirements of the POMALYST REMS™ program * Disease free of prior malignancies for \> or = 2 years with exception of treated basal cell or squamous cell carcinoma of the skin. Carcinoma "in situ" of the cervix or breast, or low-risk localized prostate cancer does not outright exclude patients, but such cases need to be discussed with Dr. Zonder prior to enrollment. * Females of childbearing potential (FCBP) must have negative serum or urine pregnancy test with a sensitivity of at least 25 mIU/mL within 10-14 days prior to and again within 24 hours of starting pomalidomide and must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before she starts taking pomalidomide; FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP even if they have had a vasectomy; all patients must be counseled at a minimum of every 28 days about pregnancy precautions and risks of fetal exposure * Able to take aspirin (ASA;81 or 325 mg) daily as prophylactic anticoagulation (patients intolerant to ASA may use warfarin or low molecular weight heparin), unless baseline prothrombin time \[PT\] or partial thromboplastin time \[PTT\] is \>= 1.5 ULN, in which case thromboprophylaxis not required Exclusion Criteria: * Any serious medical condition, laboratory abnormality, or psychiatric illness that would prevent the subject from signing the informed consent form * Pregnant or breast feeding females; (lactating females must agree not to breast feed while taking pomalidomide) * Any condition, including the presence of laboratory abnormalities, which places the subject at unacceptable risk if he/she were to participate in the study or confounds the ability to interpret data from the study * Use of any other experimental drug or therapy within 28 days of baseline * Known hypersensitivity reaction or history of desquamating rash related to thalidomide or lenalidomide * Known hypersensitivity to bortezomib, boron, or any of the other agents utilized in this protocol * Patient has \>= grade 3 peripheral sensory neuropathy or \>= grade 2 painful sensory neuropathy within 14 days before enrollment; (NOTE: patient with peripheral neuropathy \[PN\] that was previously this severe but is currently improved due to ongoing therapy \[e.g., gabapentin or amitriptyline\] may be eligible) * Myocardial infarction within 6 months prior to enrollment or has New York Heart Association (NYHA) class III or IV heart failure, uncontrolled angina, severe uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities (not including 1st degree atrioventricular (AV)-block, Wenckebach type 2nd degree heart block, or left bundle branch block; prior to study entry, any electrocardiogram (ECG) abnormality at screening has to be documented by the investigator as not medically relevant); note: there is no lower limit of left ventricular ejection fraction below which patients are excluded from participation * Concurrent use of other anti-cancer agents or treatments * Known positive for human immunodeficiency virus (HIV) or infectious hepatitis, type A, B or C * Meets criteria for symptomatic multiple myeloma, defined as: * \>= 10% monoclonal plasma cells in the marrow AND ANY OF THE FOLLOWING: * Biopsy-confirmed plasmacytoma * Lytic bone lesion(s) * Hypercalcemia without other explanation