This paper presents a novel 2.4GHz Chebyshev bandpass filter that exhibits low insertion loss in the passband and high harmonic suppression in ultra-wide stopband. Asymmetric compact microstrip resonant cells combined with a shunt open-circuited microstrip line are employed as admittance inverters instead of the traditional quarter-wavelength lines to suppress the second, third and fourth harmonics in the upper bands. Folded lines are used to realize the half-wavelength shunt open-circuited stubs to achieve a size reduction and performance improvement. By introducing a new optimization strategy, the filter can be efficiently optimized by separately optimizing the resonant cells and folded lines. Measured results show very good performance with a low insertion loss of 0.69dB and return loss of 30.5dB at the center frequency with 0.01dB equal-ripple bandwidth of 5%. The ultra-wide stopband is from 3.37GHz to 9.88GHz, where harmonics up to the fourth order are highly suppressed.