We study the properties of dissipative solitons supported by a chirped lattice in a focusing Kerr medium with nonlinear loss and a transversal linear gain landscape consisting of single or three amplifying channels. The existence and stability of two types of dissipative solitons, including fundamental and three-peaked twisted solitons, have been explored. Stable fundamental solitons can only be found in a single-channel gain chirped lattice, and stable three-peaked twisted solitons can only be obtained in a three-channel gain chirped lattice. The instability of two types of dissipative solitons can be suppressed at a high chirp rate. Also, robust fundamental and three-peaked twisted nonlinear states can be obtained by excitation of Gaussian beams of arbitrary width in corresponding characteristic structures.