I'm an immigrant and I've not even lived here a year, so I can't say the majority of the people here pronounce the g, but I asked my boyfriend (who was born and raised here) and he said it's just a matter of dialect. It's a pretty soft g from what I've heard, but it's definitely there.
Edit: I think the most important thing with dialects and pronunciation is knowing the multiple forms of pronunciation. Since you speak Swedish, you'd of course know what someone meant if they pronounced the g in 'god morgon' even if it isn't how you say it. For those of us learning Swedish, it's important to learn not only the most common/correct pronunciation but also other dialects and even the more colloquial pronunciations (a great example is all of the numbers from 20-29!). I have friends studying Swedish alongside me and since our teacher only uses the /sh/ pronunciation of sj and many of them have little interaction with other native Swedish speakers (the course is for exchange students), they have no clue what anyone in Stockholm is saying when they use the more common (in this area) /ɧ/. Those who do recognize it see it as "the way Tori pronounces it" instead of another version. One of my pet peeves of the course really :)