Martin Midstream Partners (MMLP) 10-K 2014 FY Annual report

Exhibit 23.1

Consent of Independent Registered Public Accounting Firm

The Board of Directors

Martin Midstream GP LLC:

We consent to the incorporation by reference in the registration statements on Form S-3 (No. 333-193715 and No. 333-183481) and on Form S-8 (No. 333-140152) of Martin Midstream Partners L.P. of our reports dated March 2, 2015, with respect to the consolidated balance sheets of Martin Midstream Partners L.P. and subsidiaries as of December 31, 2014 and 2013, and the related consolidated statements of operations, comprehensive income, changes in capital, and cash flows for each of the years in the three-year period ended December 31, 2014, and the effectiveness of internal control over financial reporting as of December 31, 2014, which reports appear in the December 31, 2014 annual report on Form 10-K of Martin Midstream Partners L.P.

/s/ KPMG LLP

Dallas, Texas

March 2, 2015