A funny story I tell from time to time, is that I was with a company as the CTO that was acquired by a huge conglomerate with headquarters in Spain, so we had a lot of Spaniards that would come over to work for 6 month to several years. Anyways, one of them, Valentine was the CTO and my Spanyard counterpart. Soon this game began among several of the Spaniard executives to see who could learn obscure US colloquialisms and use them in conversation. That being said, the south has a master on them in spades. So me and one of my senior developers set about to teaching Valantine southern colloquialisms. Which would sometimes come out of his mouth hilariously wrong. Anyways, Valentine and myself end up late to a meeting one day and some decision were made that neither of us agreed with and Valentine says listen you carpetbagging hornswogglers, that dog is not going to hunt (translation, you untrustworthy, scheming people, that idea is stupid and it's not going to work). You could see the absolute defeat on the other Spaniard execs faces. It was a hilarious interaction, in which the other execs where trying to get me to cough up some colloquialisms they could use.
I never suppressed my accent in my career and have done well for myself. That being said, there have been occasions where I could tell I was in a room full of people who assumed I was a dumb southerner. That was to their detriment not mine.
As a side note, I did teach myself how to speak in a completely mid-atlantic accent which is pretty cool because it's not a real accent so nobody knows where you are from. If you want to teach yourself an accent, I would suggest learning mid-atlantic, as you will never run across a native speaker, it will throw people off and strike up a lot of conversation.
It's funny but accents, at least in native language are actually fairly fungible. If someone takes the effort, they can easily fake an accent to fairly realistic abilities so long as they are familiar with the dialect. I can fake a Italian/Brooklyn accent to fairly decent accuracy as most native English speaking Americans are able to. It might not fool a Brooklyn native, but it is good enough to fool the average American. We can thank Rocky Balboa for that. I can fake most southern accents to the extent that I can fool native speakers, but unless you are intricately familiar with the region it is hard to fool natives because there are usually hyper-dialects, I can tell where a person is from if they are from the south based on the southern accent. Yet, most people out of the south cannot tell the difference. For example, a Texas draw from the Old Florida draw, I can tell because I am from Florida, I have it and I get asked if I am from Texas all the time, yet I know when someone is from Texas as it is different in it's inflections. As well Georgia and Alabama are distinct from North Carolina and Tennessee wheras Mississippi is a mix, get too far north and they sound like Tennessee, east and it sounds like Alabama, south west and it sounds like Cajun, but it blends and that is how you can tell they are from Mississippi because they will use colloquialisms from other local accents.