The only thing better than being unique
This is on top of @tim's excellent review.
I think this is a very good solution,
and a clever use of LinkedHashMap.
I can't think of a better or simpler algorithm.

Simplifications

This can be written simpler:


if (letters.containsKey(current)) {
    letters.put(current, false);
} else {
    letters.put(current, true);
}



Like this:

for (int i = 0; i &lt; input.length(); i++) {
    char current = input.charAt(i);
    letters.put(current, !letters.containsKey(current));
}


Instead of comparing boolean expressions to true like this:


if (entry.getValue() == true) {



You can use them directly:

if (entry.getValue()) {


Unit tests

It's nice that you start adding unit tests. It will pay off!

To run your test cases in an IDE,
you need to make them public (at least I had to, in IntelliJ).

As a small extra tip,
to avoid long repeated calls like NoRepeat.retrieveFirstNonRepeatedLetter(...),
sometimes it can be practical to create a helper method, for example:

void assertFirstNonRepeatedLetter(char expected, String string) {
    assertEquals(expected, NoRepeat.retrieveFirstNonRepeatedLetter(string));
}

@Test
public void toothyTest() {
    assertFirstNonRepeatedLetter('h', "tooth"); 
}

@Test
public void smellyTest() {
    assertFirstNonRepeatedLetter('d', "odor"); 
}


Why fall back to firstNonRepeated = ' ' ?

I suggest to pay close attention to the specs.
It doesn't mention what should happen if there are no unique letters in the input.
Don't assume arbitrary fallback values.
The behavior is undefined, so it's kind of an illegal state.
It's a good move to throw an IllegalStateException in this case:

public static char retrieveFirstNonRepeatedLetter(String input) {
    Map&lt;Character, Boolean&gt; letters = new LinkedHashMap&lt;&gt;();

    for (int i = 0; i &lt; input.length(); i++) {
        char current = input.charAt(i);
        letters.put(current, !letters.containsKey(current));
    }

    for (Map.Entry&lt;Character, Boolean&gt; entry : letters.entrySet()) {
        if (entry.getValue()) {
            return entry.getKey();
        }
    }

    throw new IllegalStateException("No non-repeated characters");
}


And add a corresponding unit test to cover this:

@Test(expected = IllegalStateException.class)
public void nothingUnique() {
    NoRepeat.retrieveFirstNonRepeatedLetter("abab");
}