Exhibit 32.1

Certification by Chief Executive Officer pursuant to

18 U.S.C. Section 1350,

as adopted pursuant to

Section 906 of the Sarbanes-Oxley Act of 2002

In connection with the Annual Report of Cree, Inc. (the “Company”) on Form 10-K for the year ending June 25, 2006 as filed with the Securities and Exchange Commission on the date hereof (the “Report”), I, Charles M. Swoboda, Chairman, Chief Executive Officer and President of the Company, certify, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, to my knowledge, that:

(1) The Report fully complies with the requirements of section 13(a) or 15(d) of the Securities Exchange Act of 1934, as amended; and

(2) The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.

/s/    CHARLES M. SWOBODA

Charles M. Swoboda

Chairman, Chief Executive Officer and President

August 24, 2006

This Certification is being furnished solely to accompany the Report pursuant to 18 U.S.C. § 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, and shall not be deemed “filed” by the Company for purposes of Section 18 of the Securities Exchange Act of 1934, as amended, and shall not be incorporated by reference into any filing of the Company under the Securities Act of 1933, as amended, or the Securities Exchange Act of 1934, as amended, whether made before or after the date of this Report, irrespective of any general incorporation language contained in such filing.

A signed original of this written statement required by Section 906 of the Sarbanes-Oxley Act of 2002 has been provided to the Company and will be retained by the Company and furnished to the Securities and Exchange Commission or its staff upon request.