In a British accent:

Would you fancy taking a leisurely a stroll in the park with me my Dear!?

In New York accent:

How's bout ya get up and try walkin' in the park w'me you lazy good fer nuthin?

In Texas accent:

Butter my butt and call me a biscuit, it's less than 100 degrees outside! I'm feeling like a little fresh air would do us some good so how's bout y'all tie up them dawgs and we giddy up and take a round bout the ranch some?

Hee heee. 

I've never heard the word "fancy" used as a verb in General American English.  I really fancy my wife, but I would never tell her that I fancy a trip to the park with her.  :)    

I see, you're learning Russian, so I added a few comments.

Пошли в...?  (usage of past tense in the meaning of 'let's...' is unique to this verb)
Пойдём в...?
Давай (let's) пойдем в...? (давай пошли would be incorrect)
Давай сходим в...?
А не пойти/сходить ли нам в...? 
(synonims for 'go/walk' is a long story: пройтись, прогуляться..)
Может (maybe) пойдём в...?

Хочешь (do you want?) сходить/сходим в...?
Не хочешь (don't you want)...?

These are met in fiction books (especially in translated ones) more often than in everiday speech, but still they make sense:

How about we go to the park? - Как насчет + a noun...? Как насчет рогулки в парк?
How do you feel about... Что ты думаешь (think) насчет...

why don't we go.... Почему бы нам не сходить? (cf. 'не сходить ли нам?')

----
айда в парк! (folk/children language)

Contractions are possible too:
Давай в парк! Lets to park
Может в парк? Maybe to park?
Хочешь в парк? Не хочешь в парк?
----
Сходи со мной в парк? (just 'go to the park with me':))))

'А' can be added to any of above:
A пошли в парк?
А не хочешь сходить в парк?
А сходи со мной в парк?
Literally in means 'and' 'on the contrary', 'by the way' - consult the dictionary for Russian 'a'.
It may add either enthusiasm or uncertainty:)

If I were to ask my wife about going to the park with me, then I would ask her "Do ya wanna go to the park?"  :)