The only downside to watching the Naruto anime in order is that you have to deal with a lot of filler. Feel free to skip the filler episodes, but you should at least consider watching these ones.

Naruto: Episodes 97, 101, 145, 148, 151, 157, 206 

Naruto Shippuden: Episodes 57-71, 194, 290-296

Ultimately, watch Naruto how you want to watch it. It may be difficult to watch the episodes out of order, but the OVAs and movies can be watched as a stand-alone thing — though I personally wouldn't recommend it. 

As with all good fictional worlds, your appreciation of characters and plot deepens when you follow them in a certain chronology.