I'm an Independent Voter - Can I Vote in the Primary in Kansas, Illinois, Connecticut, or Arizona??

KANSAS -- Yes you can!
From the De Soto Explorer:
Voters affiliated with a political party who want to participate in another party’s primary must change their affiliation by Monday. But independent voters can change their affiliation on Election Day. Independent voters can also participate in the Democratic primary without changing their affiliation.

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ILLINOIS -- Uhhh, well yes, but then you become one of them, but then you can unaffiliate... but then everyone knows what ballot you chose...

From FairVote

"Must vote in primary of same party as last primary the voter participated in. Loosely enforced.  Voters may change party affiliation at polls or caucus." 

The big hoopla going on right now in Illinois re: the gov race is that voters feel their jobs can be threatened if they don't vote in the primary according to the party in power. Gov Pat Quinn just used his veto power to set up an open primary (we'll see what the legislature does with that). Quinn's independent challenger Scott Cohen also supports open primaries. 

Chicago Tribune's Eric Zorn says: "Under Illinois' system, voters effectively register their affiliations when they ask for party ballots. But either way, the information becomes known to the world — or at least to the party operatives who use it to target direct-mail campaigns, get-out-the-vote efforts and so on."  Read more Zorn here.

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CONNECTICUT -- No!

From Damon Eris at Poli-Tea

Connecticut has closed primaries. Thus, almost half of all registered voters will be ineligible to vote in the major party primaries, whose nominees will therefore be decided by a fraction of a fraction of a fraction of registered voters.... 

Rather than open the primary system, however, Connecticut's political establishment has instead chosen to begin brow beating voters to register with the Democratic and Republican parties. Late last week, Democratic Secretary of State Susan Bysiewicz held a press conference (.pdf) with leaders of the state's Democratic and Republican party establishment to demand that voters give up their independent affiliations and enroll in the Democratic and Republican parties ahead of the primaries. From the press release:

Secretary of the State Susan Bysiewicz today urged unaffiliated voters across the state to enroll with the Democrat or Republican Party by August 9th at 12:00 p.m. so they can vote in the statewide primaries Tuesday August 10, 2010. Connecticut's closed party system does not allow unaffiliated voters to cast ballots in the primaries, therefore unaffiliated voters must enroll as either Democrats or Republicans to cast a ballot in the primaries....

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ARIZONA -- Yes you can (at the state level)!

From Arizona Republic

In Arizona, independent voters can choose to participate in a partisan primary.... One of three Arizona voters is registered as independent. That large swath of Arizonans shouldn't be silent when their voices can make a difference. Unfortunately, many independent voters mistakenly believe that they're shut out of party primaries. They used to be. But 12 years ago, Arizona voters approved an open-primary system that lets independents participate in all but presidential primaries. (Parties set up their own presidential-preference systems.)
NOTE: Arizona independents are not allowed to vote in partisan primaries for President.
From OpenPrimaries.org
In February, Democratic Party legislators introduced HB 2579, which provides that a voter who is registered as independent, or as “no party preference,” may vote in any eligible party’s presidential primary, unless that party tells the Secretary of State otherwise at least 85 days in advance of the primary.