Provoke in a good way

Solution 1:

Inspire and encourage are clearly positive, but not as strong.

Provoke is generally negative as you say, though if used of a good thing then it could work well, precisely because it would be a use that is unusual—whether it's so unusual to jar or just unusual enough to stand out well can't be judged out of context, and there would likely be some disagreement if we did see the full sentence.

Spur is perhaps a good choice. Edit: some people are objecting to spur, they've a fair point, but I hope none of them ever ride horses ;)

Incite, foster, hearten, embolden, induce, stimulate, persuade, coax, instigate, and urge are all words that overlap enough that they might suit your purpose but not so much that they necessarily would.

Edit: Stir works in some cases too, and while it's a bit old-fashioned in this sense, I personally like it.

Solution 2:

"arouse" comes to mind also. Though, be careful when using, it can be too positive sometimes.

Solution 3:

Words that come to mind when thinking of "provoking" but with a positive connotation would be:

motivating
prompting
driving
encouraging
incentivizing

Solution 4:

How about challenge? To challenge someone to to something. It seems to me it fits the bill for positive provocation.

Solution 5:

How about the the word "rouse"?

rouse: to kindle to intensity : excite, inflame

http://www.merriam-webster.com/dictionary/rouse