Please don't listen to these users. The only difference between inline and external css is that it will get cached with external, but not inline. So it will certainly not go any faster to use inline instead. In fact, it will even go slower with external, as this is an extra http request. However, these micro-optimizations will certainly not make a difference, at least in terms of SEO.
That said, it is better to use one external css stylesheet for caching purposes. Additionally, inline css tends to be very cluttered and unreadable. I'd suggest doing internal css over inline though.