Well, to know the effects, you'd really need the details for the new system. For example, who certifies the vote? Who does the counting? Are their deadlines for certifying? And the biggy: what's the threshold for victory? A plurality? A full majority, if so what's the provision for a recount?
It will have some profound effects, particularly in places like South Carolina, where the State Legislature still directly voted on presidential elections. Abolishing the electoral college is essentially forcing democratization down their throats.
The bigger effects are to the system of party organization, which was in its infancy in the 1830s. The electoral college is a large part of the impetus that keeps the American system a two-party one. True, you'd still have first-past-the-post in most of Congress, but without the hope of influencing presidential elections (or rather nominations), there's a lot more power to be had by third parties, particularly if they can get into Congress and/or state legislatures. TTL this incentive is broken at a crucial time in American political development, especially if the presidency goes to a plurality winner, rather than to the winner of a simple majority. And this will only compound the political chaos of the period. OTL the Whigs' collapse, the rise of the Free Soil and then Republican Parties betoken a political realignment which encourage sectional division. The electoral college and presidential contests premised on it were essential to this story. I won't go so far as to say that it will prevent/accelerate the comming of the Civil War, but it will certainly change things in a major way.
Oh, one more thought: the south is going to have a very, very hard time swallowing a popular vote, since with the popular vote, the 3/5s clause loses some its teeth. Jackson of course had the popularity in at least the backwoods part of the South necessary to overcome this objection, but it will make Southerners even more fearful.