In Indian English, the phrase "Tom, Dick and Harry" is very prevalent, is used quite frequently in media, movies etc.
On the other hand, I don't see it being used as frequently, in American media.
What phrase would an American use in its place?
Context:
Any tom, dick and harry can solve this problem.
Used to refer to ordinary people in general
As Michael Siefert says, this is both used and immediately recognizable in American English, but it would be generational, with the oldest speakers finding it natural, many or most middle-aged speakers finding it antiquated, and younger speakers likely being unfamiliar with it. The reason you are not seeing it in media is because the mass-media would focus on the middle demographic and so would only use it when they wanted to imply old-fashionedness.
Youth typically use 'literally' as an emphasizer, so young people would say "Literally anyone could solve this". However, that would be too slang to be used in mass media (although common in demographically targeted social media).
More recognizable alternatives include "Any Joe (or any regular Joe or any Joe Shmoe)," "any guy off the street", "any idiot", "any fool", and "anyone with half a brain". The first two emphasize how easy the problem is such that any common person has the ability to solve it, and the second three that even someone of limited ability would be able to solve it.
Other answers and comments have suggested 'everyone and their brother' or 'everyone and their dog'. I think there is a subtle distinction here between anyone and everyone, that is, between saying that the problem is so easy that a single person of the lowest common denominator could solve it, and saying it is so easy that every person in any given group would be able to solve it. I think this is the same distinction that Darrel Hoffman and gotube are drawing between 'Tom, Dick, or Harry' and 'Tom, Dick, and Harry', but I don't know if that distinction is used in Indian English or what the OP's original intent was.
Since I have just finished my yearly re-watch of GATTACA (I show it to my students) I feel I should include a quote from Jerome / Eugene:
That was the company that sell us your hair dye. They sent me ''summer wheat" instead of "honey dawn." Any fool knows it's two shades lighter.
A common casual phrase is "everyone and their brother". YourDictionary has the following definition:
(idiomatic) A large number of people; most people.
Similar phrases are "everyone and their dog" and "everyone and their mom". The general idea of all of these is that it doesn't require any special expertise -- "everyone" is interpreted as people familiar with the activity in question, but "their brother/mom/dog" would not be assumed to have similar experience. The variant with "dog" is especially broad, since most dogs can't do anything that humans are especially good at (but they have their own abilities that make up for it).
One could simply say
Anybody/anyone could solve this problem.
As "tom, dick and harry" refers to an ordinary person in general. Usually in most cases, "anybody/anyone" is used to describe the general public/people.
I would suggest that a similar phrase would be: Average Joe
An excerpt from Wikipedia's page on the term includes:
used primarily in North America to refer to a completely average person, typically an average American. It can be used both to give the image of a hypothetical "completely average person"
In the example given in your question, it would be used like:
The average Joe can solve this problem.
Or some similar phrasings:
Your average Joe can solve this problem.
Any average Joe can solve this problem.
Kirt mentions "Joe" in their answer, but I think that specifically an average Joe is equivalent.
"Average Joe" is the first item listed in the "See also" section of the Wikipedia page for "Tom, Dick, and Harry".
"Average Joe" is on this list of terms referring to an average person, on which is also found "Tom, Dick, and Harry": https://en.wikipedia.org/wiki/List_of_terms_referring_to_an_average_person
It is native English, so it would be perfectly understandable as is, although or is used more frequently. It just means "anyone". It can be used with "any old" as well (which is the form familiar to me).
Any Tom, Dick and Harry [unexpected people] can walk across my lawn.
I'll just get any old Tom, Dick or Harry [anyone will do] to paint the shed.
It is considered outdated, but actually increasing in usage since about 2000. The Google Ngram viewer illustrates this, although note the form with commas cannot be searched due to limitations in the crude nature of that company's "advanced" search features.
They'd probably say; "any old Joe".
There's an interesting usage of that, showing (IMHO) what to make of it, nowadays:
In a translation from Chaucer's "Canterbury tales" (middle English → modern English), we read in the original of the friar's tale:
He had eke wenches at his retinue, also prostitutes
That whether that Sir Robert or Sir Hugh,
Or Jack or Ralph or whoso that it were
That lay by them, they told it in his ear.
The translation says
Moreover, he had harlots at command, and so, if Reverend Robert, or Reverend Hugh, Tom, Dick, or Harry, or whoever, had one of them, it reached the summoner's ear".
So it just means an arbitrary list of names, potential customers of "wenches"="harlots". And, being used as a translation of this ancient tale, it carries the notion of "a bit ancient".
