I want to get from TLV to Thessaloniki, Greece on 21-24/June.
What is the cheapest way?
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You can try a boat, they used to have pretty cheap tickets on ships between Greece and Israel a decade or two ago, but I'm afraid its not as popular as it used to be, so you might not find any.
Sea and air are your only options, unfortunately.
As overland travel, either by hitchhiking or by public transport (bus+train), is not really an option at the moment, flying will probably be the cheapest option.
On kayak, I have seen direct one-way flights from Tel-Aviv to Tessaloniki for 576 NIS (~120 EUR).
Going by ferry (if you find one) will hardly be cheaper. In 2011, there used to be a ferry service between Israel and Cyprus. The posted price is 250 EUR, one-way. From Cyprus, it is possible to take a ferry to the Greek mainland.