Introduction. Let ffi be an adjoint, simple algebraic group of type E,(n = 6, 7, 8), F 4 or G2 over an algebraically closed field K of characteristic p > 0. We assume that (5 has a fixed Fq-rational structure for a finite subfield Fq of K. The purpose of this paper is to determine explicitly the values of the irreducible characters of G = ffi(Fq) at the unipotent elements when p is good [38; I, 4.3]. In this Part I, we shall do this only for large p, and the full result will be proved in the forthcoming Part II. (After submitting the first version of the present paper, the author was informed by Lusztig that he solved analogous problems for classical groups using his theory of character sheaves, and that he can also recover the result of this paper mentioned above.) Our method depends upon: (1) a parametrization of the irreducible characters of G and the determination of their multiplicities in Deligne-Lusztig virtual representations (Lusztig [25]); (2) the computation of the Green functions of G in large characteristic (Springer [36; 7.1@ Shoji [33], Beynon and Spaltenstein [3, 4]); (3) the determination of the values of generalized Gelfand-Graev characters [20] of G in good characteristic (see Section 3). By (1) and (2), the "uniform parts" of the irreducible characters are already known explicitly in large characteristic. This implies that, in large characteristic, our problem for groups of type E 6 o r E 7 has already been solved. (It should be mentioned here that the whole character table of the G2-group is also known by Chang and Ree [8]. Hence, in the G2-case, this paper gives just another approach to (a part of) the result of Chang and Ree.) If G is of type G2, F4 or E 8, the restriction of an irreducible character r of G to the set of unipotent elements can be written as a sum of its uniform part with a function of the form