This paper proposes a near-capacity yet simple bit-interleaved coded modulation with iterative decoding (BICM-ID) scheme by employing a multi-dimensional (multi-D) mapping technique in a multi-D constellation carved from a rotated lattice. Using extrinsic information transfer (EXIT) charts, it is shown that the proposed technique fits well with simple convolutional codes in terms of the area property, for which turbo pinch-off can happen at a low Eb/N0 value. In particular, both EXIT chart analysis and simulation results indicate that by using just a simple convolutional code together with a 4-D mapping, a turbo pinch-off and a bit error rate (BER) close to 10-6 happen at a signal-to-noise ratio (SNR) that is even lower than the BICM constraint capacity limit with a uniform input. The proposed BICM-ID scheme can be considered as an attractive alternative to other bandwidth-efficient coded modulation techniques using powerful turbo-like codes such as turbo or low-density parity-check (LDPC) codes over a Rayleigh fading channel.