It started as a comment, but it grew into an answer instead.
Typical for the Hessian dialect (not accent) is the shift from /ʃ/ and /ç/ to /ʒ/ as illustrated in Martin Schneider's Aschenbecher example.
Within the Hessian dialect, it's also quite common to skip the n in an -en ending and put more emphasis on the (now) final e. Again, this can be heard in the Aschenbecher example above and also be observerd in the examples below
Numerous examples are listed in this German online dictionary, where some heuristic rules can already be derived.
auf tends to become uff
p tends to become b
Babba (Papa), uffbasse (aufpassen)
-ig tends to become -isch
hibbelisch (hibbelig), ferdisch (fertig)
ck tends to become gg
aijereggisch (eiereckig), hogge (hocken), Schnuggelsche (Schnuckelchen)
In summary, a lot of hard sounds become softer.
What are specific words, only used in Hessen
Iconic Hessian words are
Aijereggisch, Riwwekuche (Streuselkuchen), babbeln, Bembel, Gude (Begrüßung), (Hut-)simbel, ...
Meeting older people from Hesse or people who live/grew up in smaller villages are still speaking the dialect, especially towards the south around the Taunus area.
This is mainly based on my observations as a North-German living in Hessen and noticing the differences with regard to my own dialect.