Well, maybe you would like Mignarda, an American man and wife team, who have dedicated themselves to this repertoire for many years now:
http://www.mignarda.com
I too dislike the operatic voice for this repertoire, but that does not mean a pop singer approach works any better. I don't know where to begin with the lute playing, which has far more in connection with contemporary steel-strung guitar or modern classical guitar, than anything John Dowland would have recognised. The two lutes Sting and his friend play, were completely unknown in Dowland's day, and were constructed for music with a different aesthetic.
Either way, operatic or pop, neither gets close to Dowland. This, in my humble opinion, does:
It is not surprising that many people like what Sting does with these songs - he is a great communicator, and I've almost always liked his work. But I just can't bear listening to his Dowland performances. It's not a case of my being a musical snob, it's more a case of the colouring they bring to Dowland's music is so at odds with the musical and poetic nuances of the Renaissance, that what they end up with is something new - nothing wrong with that - but I'm more interested in Dowland than in Sting.
The reason I gave a short reply to Tom is that I much prefer talking about what I like, rather than what I don't like - there is far too much of that going on on the internet already. Listen to the nuances the Mignarda couple draw from these songs (rather than impose on the songs). There is a self-effacing beauty there that is also very personal - a typical Renaissance conceit - where the cult of the singer or the lute player has no place.
We may never agree on this one, which is also why I didn't want to discuss it