That really isn't true, or at least it isn't true with some practice. You don't have to consciously think about or learn tones, but you will eventually pick them anyways (tones are learned unconsciously via lots of practice trying to speak and be understood).

You can be perfectly understood if you don't speak broadcast Chinese. There are plenty of heavy accents to deal with anyways. Like Beijing 儿化 or the inability of southerners to pronounce sh very differently from s.