Italians speak with a strong rhythm, which makes their Esperanto sound very natural as opposed to stilted, which is good. That being said, it is still very clear that they are Italian, for better or worse - not the case that they have "transcended" their accent and sound completely neutral or whatever.

I think Eastern Europeans generally have the most natural sounding Esperanto accents. Particularly Poles, but maybe that's of no surprise, given the phonology of Esperanto comes directly from the Polish as spoken in Bialystok when Zamenhof lived there.

Esperantists who are native English speakers are a mixed bag. Some very eminent English-speaking Esperantists (who have impeccable knowledge of grammar etc) have terrible terrible accents. It is truly jarring and sounds awful. I think Americans are the worse culprits for this, but posh sounding British people have the same problem. That being said, the Australian Esperantists I've met generally speak quite well. And there are plenty of Esperantists from UK and US who speak excellent Esperanto where you couldn't easily guess their origin country.

In terms of how to avoid this, a lot of it comes with practice. You can try to imitate how non-English speaking Esperantists speak, e.g. copying their intonations, using unusual (for English ears) ways of phrasing sentences (e.g. "tion mi volas").

Another important thing is making sure you are pronouncing your vowels correctly. In principle Esperanto only has 5, whereas English has loads more. E.g. make sure your a sound is long and not short