Power-domain asynchronous non-orthogonal multiple access (ANOMA) is a novel radio access technique with non-orthogonal resource allocation that enables asynchronous transmissions and has an enhanced spectrum efficiency compared to orthogonal multiple access. In this work, an iterative receiver is derived for linearly modulated waveforms. Orthogonal frequency division multiplexing (OFDM), single-carrier (SC) and orthogonal chirp division multiplexing (OCDM) are investigated. The receiver is based on triangular successive interference cancellation (T-SIC) in combination with a minimum mean square error parallel interference cancellation (MMSE-PIC) detector. It is advantageous to utilize a waveform which spreads the data symbols in the frequency domain as OCDM or SC in order to exploit the multipath diversity in frequency-selective channels. However, it is numerically shown that OCDM performs the best due to its additional time-spreading property, which is desirable for the time-dependent interference that occurs in an ANOMA system. Furthermore, for the considered scenario of two users and four blocks, we show that all the studied waveforms achieve the best performance in terms of block error rate with the derived receiver when the blocks overlap halfway.