The popular image of a Civil War battlefield is empty fields or rolling hills without underbrush or many other obstacles. The combatants can see one another from a reasonable distance and organizing stereotypical 19th century formations is simple. But this doesn't describe much of the Mid-Atlantic or Southern United States today, much less in the 1860s.
So was that actually how most Civil War battlefields were? Or is our popular image missing a lot of the color of the past?
A couple things come into play with this question, I think: visual representations of battle in the American Civil War, and the nature of large-scale 19th century combat.
Starting with historical examples of 19th century combat, specifically American Civil War battles, the answer is no: most battles did not occur on wide open plains or soft rolling hills with an absence of obstacles. There may have been parts of the battlefield like that, but armies of 20,000, 30,000, or even 50,000+ men meeting bodies of a similar size on an open plain that could fit all of them was exceedingly rare throughout the conflict (and I'd dare say, throughout the course of human history). In any large scale battle, there was likely several different minor theaters of action, and many of those dealt with different terrain and obstacle challenges.
Let's look at some examples to illustrate. Antietam is often examined as three smaller battles over the course of one day and area, with separate fights along the sunken road ("bloody lane"), the cornfield, and the "Burnside's Bridge" over Antietam Creek. All of these locations had some degree of "openness" though what is remembered about each within the context of examining the flow and ebb of battle revolves around the obstructions and obstacles. Gettysburg is notable for Pickett's Charge on the 3rd day, which did span a wide-open plain with just one snake rail fence obstacle (and a low stone wall in front of Union lines), but much of that battle took place on wooded hills, rocky outcrops, and even an orchard. Likewise, although the Battle of Fredericksburg is often remembered for the bloody assaults on Marye's Heights, which was indeed a mostly open space without obstacles, most of the other portions of that battle occurred in wooded areas or across a river.
In fact, most commanders did whatever they could to avoid open field fighting, preferring instead to steal a march on their opponent (getting between their base of operations or supply lines) or force them to attack from an exposed position (much like an open field). Although it did happen on occasion on a grand scale (the Battle of Franklin) it usually spelled disaster.
So yeah, there were open field battles with little or no obstacles, but that was the exception not the norm. It was not good tactics, and unless one had an overwhelming advantage in numbers, such a proposition was doomed to fail for the attacker. Now, as to why Civil War combat is sometimes portrayed in this way, the simple answer is that such a visual spectacle is far easier to process for an observer. This wood carving from the period https://www.loc.gov/resource/cph.3b45350/ can be found at the Library of Congress. The top left panel shows an image from Gettysburg with the wide open plains in the foreground (with hills far away in the distance). This doesn't conform to any real written account of the battle in any place (Pickett's charge didn't really look like this), yet it is a striking image that conveys the scale and scope of the engagement in a way that "reads" easily to anyone, literate or not.
What we now might understand as a stereotypical Civil War battle in terms of visuals conveying a wide-open plain with clearly defined lines and formations is more of a cinematic development. You might say that 19th century wood carvings and lithographs were more accurate in their portrayal of Civil War action than we are today with movies (this one seems entirely appropriate, as do many others: https://www.loc.gov/item/00652060/). Many pieces of art from this period do represent obstacles like wooded thickets, streams, hills, etc. A lot of Civil War movies feature the bigger, open set-pieces, however, because again, the visual conveys more in a shorter amount of time. If a filmmaker is going to film a battle, they aren't going to obscure the screen with smoke and debris and obstacles unless they have to (or unless the point of the scene is to present this to the audience).
For example, in the film Gettysburg, the Battle of Little Round Top does take place in a wooded area and up a hillside like the actual battle, but the grand set-piece where most of the budget went was Pickett's Charge. In Gods and Generals (2003), the biggest set pieces were Fredericksburg and Chancellorsville, and while neither were exclusively open, that's what gets the most screen-time because it is admittedly easy to process visually (clear lines, easy demarcation between sides, etc). Cold Mountain (2003) is another example of a very complicated, messy battle (Battle of the Crater) without an open field as OP mentioned, but visually, somewhat clean lines between sides for the first portion (until the scene very accurately descends into abject carnage and chaos).
TD;LR - No, the majority of Civil War battles were not fought exclusively on open fields with clear sight lines. Some art from the period did depict it this way, though not all. The modern portrayal of this phenomenon is due to the strictures of filmmaking and visual communication with audiences.