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Step 1: Set up the number for the long division method.
Group the digits in pairs from the right, placing a bar over each pair. For 1369, we have $\overline{13} \overline{69}$.

Step 2: Find the largest number whose square is less than or equal to the first group (13).
The largest square less than or equal to 13 is $3^2 = 9$.
Write 3 as the quotient and the divisor. Subtract 9 from 13.

$$\begin{array}{r} 3 \\[-3pt] 3 \overline{| 13 \ 69} \\[-3pt] -9 \phantom{69} \\[-3pt] \hline 4 \phantom{69} \end{array}$$

Step 3: Bring down the next pair of digits (69) to the remainder (4), making it 469.
Double the current quotient (3), which is $3 \times 2 = 6$. Write 6 as the first digit of the new divisor.

Step 4: Find the next digit for the divisor and quotient.
We need to find a digit $x$ such that $6x \times x$ is less than or equal to 469.
Let's try $x=7$: $67 \times 7 = 469$.
Write 7 as the next digit in the quotient and also next to 6 in the divisor. Subtract 469 from 469.

$$\begin{array}{r} 3 \ 7 \\[-3pt] 3 \overline{| 13 \ 69} \\[-3pt] -9 \phantom{69} \\[-3pt] \hline 67 \overline{| 4 \ 69} \\[-3pt] -4 \ 69 \\[-3pt] \hline 0 \end{array}$$

Step 5: The remainder is 0, so the square root is the quotient.
The square root of 1369 is 37.

$$\sqrt{1369} = \boxed{\text{37}}$$