EXHIBIT 31.1

                    CERTIFICATION OF CHIEF EXECUTIVE OFFICER
            PURSUANT TO SECTION 302 OF THE SARBANES-OXLEY ACT OF 2002

I, Robert G. Dutcher, certify that:


1.   I have reviewed this quarterly report on Form 10-Q of Possis Medical, Inc.
     (Possis Medical);

2.   Based on my knowledge, this quarterly report does not contain any untrue
     statement of a material fact or omit to state a material fact necessary to
     make the statements made, in light of the circumstances under which such
     statements were made, not misleading with respect to the period covered by
     this quarterly report;

3.   Based on my knowledge, the financial statements, and other financial
     information included in this quarterly report, fairly present in all
     material respects the financial condition, results of operations and cash
     flows of Possis Medical as of, and for, the periods presented in this
     quarterly report;

4.   Possis Medical's other certifying officer and I are responsible for
     establishing and maintaining disclosure controls and procedures (as defined
     in Exchange Act Rules 13a-15(e) and 15d-15(e)) for Possis Medical and we
     have:

     a.   Designed such disclosure controls and procedures, or caused such
          disclosure controls and procedures to be designed under our
          supervision, to ensure that material information relating to Possis
          Medical, including its consolidated subsidiaries, is made known to us
          by others within those entities, particularly during the period in
          which this quarterly report is being prepared;

     b.   Evaluated the effectiveness of Possis Medical's disclosure controls
          and procedures and presented in this report our conclusions about the
          effectiveness of the disclosure controls and procedures, as of the end
          of the period covered by this report based on such evaluation; and

     c.   Disclosed in this report any change in Possis Medical's internal
          control over financial reporting that occurred during Possis Medical's
          most recent fiscal quarter (Possis Medical's fourth fiscal quarter in
          the case of an annual report) that has materially affected, or is
          reasonably likely to materially affect, Possis Medical's internal
          control over financial reporting; and

5.   Possis Medical's other certifying officer and I have disclosed, based on
     our most recent evaluation of internal control over financial reporting, to
     Possis Medical's auditors and the audit committee of Possis Medical's board
     of directors (or persons performing the equivalent function):


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     a.   All significant deficiencies and material weaknesses in the design or
          operation of internal control over financial reporting which are
          reasonably likely to adversely affect Possis Medical's ability to
          record, process, summarize and report financial information; and

     b.   Any fraud, whether or not material, that involves management or other
          employees who have a significant role in Possis Medical's internal
          control over financial reporting.

Date: December 9, 2004                /s/ Robert G. Dutcher
                                     --------------------------
                                          Robert G. Dutcher
                                          Chairman, President and
                                          Chief Executive Officer


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