Usually you can only book low level awards on a partner airline, and Delta is pretty stingy about releasing low level awards to Europe in the summer. For example, I've been looking at the nonstop Delta flight Seattle to Amsterdam and it is always 75,000 miles, any time of year, never 60,000. I would suggest looking for flights on Air France, sign up with their Flying Blue program to search for flights, if you can find a Classic award available you should be able to book it with Alaska miles. Availability is hard to find for summer, the airlines don't always release low level awards.
I had good luck finding tickets on AA Seattle to Rome in May. I would sign up for several credit card bonuses, if you have miles in several alliances you have a better chance of finding what you want. Diversify your miles!